Complex manifold methods in quantum field theory in curved space-time

Abstract

A real analytic space-time 𝒩 can be embedded as a real slice in a complex Riemannian manifold (M,g). A theorem due to Woodhouse [Int. J. Theor. Phys. 16, 671 (1977)], stating that real slices are necessarily totally geodesic submanifolds of (M,g), is used to find all real slices of the complexified sphere cSnr of dimension n and radius r, and of complexified Robertson–Walker space-time cℛ. The real slices 𝒩 of cSnr are n-dimensional spaces of constant curvature (±r−2) of all possible signatures. cℛ is determined by a holomorphic radius function r(x), the x-constant hypersurfaces being isometric to cSnr(x). In general, all real slices of cℛ are Robertson–Walker-type spaces (of various signatures). The interesting case that a Robertson–Walker space-time with radius function 𝓇(t) intersects a Euclidean real slice in an n-dimensional hypersurface, can only occur if 𝓇(t) is time reflection symmetric. Finally, propagators to the scalar wave operator (□–m2) on real slices 𝒩1 of cS4r are considered and their analytic continuations to other real slices 𝒩2 are discussed. There are useful continuations only if 𝒩1, 𝒩2 are isometric either to the sphere and de Sitter space-time, or to the hyperbolic plane and anti-de Sitter space-time. In this case the continuation does not depend on the relative position of 𝒩1 and 𝒩2 in cS4r. For instance, it also works if the intersection 𝒩1∩𝒩2 is one dimensional.